Symmetry breaking in two–dimensional square grids: Persistence and failure of the dimensional crossover

Dovetta, Simone; Tentarelli, Lorenzo

doi:10.1016/j.matpur.2021.12.010

“…This dimensional crossover is related to the coexistence of the one-dimensional and two-dimensional Sobolev inequalities, leading to a generalized Gagliardo-Nirenberg inequality for the doubly-periodic metric graphs. Failure of the dimensional crossover was investigated in [62] for the doubly periodic graphs with compact or non-compact defects.…”

Section: Further Directionsmentioning

confidence: 99%

Standing waves on quantum graphs

Kairzhan

¹

,

Noja

²

,

Pelinovsky

³

2022

J. Phys. A: Math. Theor.

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“…As the literature on the subject witnessed a massive growth, we refrain from overviewing it here, redirecting e.g. to [1,3,[9][10][11][12][13][14][15][16]21,26,29,32,33,[38][39][40] for some of the most recent developments and to the reviews [2,34] for more comprehensive discussions. Within the whole theory, prominent efforts have been devoted to the analysis of existence of positive standing wave solutions, with a particular focus on ground states.…”

Section: Introductionmentioning

confidence: 99%

On the notion of ground state for nonlinear Schrödinger equations on metric graphs

Coster¹,

Dovetta²,

Galant³

et al. 2023

Preprint

0

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We compare ground states for the nonlinear Schrödinger equation on metric graphs, defined as global minimizers of the action functional constrained on the Nehari manifold, and least action solutions, namely minimizers of the action among all solutions to the equation. In principle, four alternative cases may take place: ground states do exist (thus coinciding with least action solutions); ground states do not exist while least action solutions do; both ground states and least action solutions do not exist and the levels of the two minimizing problems coincide; both ground states and least action solutions do not exist and the levels of the two minimizing problems are different. We show that in the context of metric graphs all four alternatives do occur. This is accomplished by a careful analysis of doubly constrained variational problems. As a by-product, we obtain new multiplicity results for positive solutions on a wide class of noncompact metric graphs.

show abstract

On the notion of ground state for nonlinear Schrödinger equations on metric graphs

Coster

¹

,

Dovetta

²

,

Galant

³

et al. 2023

View full text Add to dashboard Cite

We compare ground states for the nonlinear Schrödinger equation on metric graphs, defined as global minimizers of the action functional constrained on the Nehari manifold, and least action solutions, namely minimizers of the action among all solutions to the equation. In principle, four alternative cases may take place: ground states do exist (thus coinciding with least action solutions); ground states do not exist while least action solutions do; both ground states and least action solutions do not exist and the levels of the two minimizing problems coincide; both ground states and least action solutions do not exist and the levels of the two minimizing problems are different. We show that in the context of metric graphs all four alternatives do occur. This is accomplished by a careful analysis of doubly constrained variational problems. As a by-product, we obtain new multiplicity results for positive solutions on a wide class of noncompact metric graphs.

show abstract

Symmetry breaking in two–dimensional square grids: Persistence and failure of the dimensional crossover

Cited by 6 publications

References 56 publications

Standing waves on quantum graphs

Standing waves on quantum graphs

On the notion of ground state for nonlinear Schrödinger equations on metric graphs

On the notion of ground state for nonlinear Schrödinger equations on metric graphs

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